Subgroups — C₂²×C₁₃×#493 #15746968

Order 3328.

4638 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×D₈	C₂	1	4	yes	maximal, complemented
2	C₂²×C₁₃×#127	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₃×#493	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₄×C₁₃	—	2	4	no	—
6	C₂³×D₄×C₁₃	C₂²	1	4	yes	—
7	C₂²×C₁₃×D₈	—	2	32	no	—
8	C₂²×C₁₃×D₈	C₂²	1	48	yes	complemented
9	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
10	C₂×C₁₃×#127	—	2	12	no	—
11	C₂×C₁₃×#127	C₂²	1	24	yes	complemented
12	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₁₃×#493	C₂²	1	64	yes	direct product factor
14	C₂×C₄×C₈×C₁₃	C₂²	1	12	yes	complemented
15	C₂⁵×C₁₃	—	2	4	no	—
16	C₂²×D₄×C₁₃	—	2	32	no	—
17	C₂²×D₄×C₁₃	—	2	12	no	—
18	C₂²×D₄×C₁₃	—	2	48	no	—
19	C₂²×D₄×C₁₃	—	4	32	no	—
20	C₂²×D₄×C₁₃	C₂³	1	24	yes	—
21	C₂³×C₄×C₁₃	D₄	1	2	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₁₃×D₈	—	2	192	no	—
24	C₂×C₁₃×D₈	C₂³	1	64	yes	complemented
25	C₂²×C₈×C₁₃	D₄	1	16	yes	complemented
26	C₂²×C₈×C₁₃	C₂³	1	12	yes	—
27	C₁₃×#127	—	2	48	no	—
28	C₁₃×#127	C₂³	1	32	yes	complemented
29	C₂×C₄²×C₁₃	D₄	1	6	yes	—
30	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
31	C₄×C₈×C₁₃	C₂³	1	16	yes	complemented
32	C₂²×#493	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂⁴×C₁₃	—	2	24	no	—
34	C₂⁴×C₁₃	—	4	16	no	—
35	C₂⁴×C₁₃	—	4	32	no	—
36	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×D₄×C₁₃	—	2	96	no	—
38	C₂×D₄×C₁₃	—	2	24	no	—
39	C₂×D₄×C₁₃	—	2	64	no	—
40	C₂×D₄×C₁₃	—	4	48	no	—
41	C₂×D₄×C₁₃	—	4	192	no	—
42	C₂×D₄×C₁₃	C₂⁴	1	16	yes	—
43	C₂²×C₄×C₁₃	D₈	1	16	yes	complemented
44	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
45	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
46	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
47	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
48	C₁₃×D₈	—	2	128	no	—
49	C₂×C₈×C₁₃	C₂×D₄	1	48	yes	complemented
50	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
51	C₄²×C₁₃	C₂×D₄	1	12	yes	—
52	C₄²×C₁₃	C₂⁴	1	4	yes	—
53	C₂³×D₈	C₂×C₁₃	1	4	yes	complemented
54	C₂²×#127	C₂×C₁₃	1	2	yes	complemented
55	C₂×#493	C₂×C₁₃	1	24	yes	direct product factor
56	C₂²×C₄×C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
57	C₂³×C₁₃	—	2	16	no	—
58	C₂³×C₁₃	—	4	48	no	—
59	C₂³×C₁₃	—	4	96	no	—
60	C₂³×C₁₃	—	8	64	no	—
61	C₂³×C₁₃	#127	1	4	yes	—
62	C₂³×C₁₃	C₂×D₈	1	8	yes	—
63	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
64	D₄×C₁₃	—	2	32	no	—
65	D₄×C₁₃	—	4	48	no	—
66	D₄×C₁₃	—	4	128	no	—
67	C₂×C₄×C₁₃	#127	1	12	yes	—
68	C₂×C₄×C₁₃	C₂×D₈	1	48	yes	complemented
69	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
70	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
71	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
72	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
73	C₈×C₁₃	C₂²×D₄	1	16	yes	complemented
74	C₂³×D₄	—	2	4	no	—
75	C₂³×D₄	C₂²×C₁₃	1	4	yes	—
76	C₂²×D₈	—	2	32	no	—
77	C₂²×D₈	C₂²×C₁₃	1	48	yes	complemented
78	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
79	C₂×#127	—	2	12	no	—
80	C₂×#127	C₂²×C₁₃	1	24	yes	complemented
81	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
82	#493	C₂²×C₁₃	1	64	yes	direct product factor
83	C₂×C₄×C₈	C₂²×C₁₃	1	12	yes	complemented
84	C₂²×C₁₃	—	4	16	no	—
85	C₂²×C₁₃	—	4	32	no	—
86	C₂²×C₁₃	—	8	96	no	—
87	C₂²×C₁₃	#493	1	16	yes	direct product factor
88	C₂²×C₁₃	C₂×#127	1	6	yes	—
89	C₂²×C₁₃	C₂²×D₈	1	12	yes	—
90	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
91	C₄×C₁₃	C₂×#127	1	6	yes	—
92	C₄×C₁₃	C₂²×D₈	1	16	yes	complemented
93	C₄×C₁₃	C₂³×D₄	1	2	yes	—
94	C₂⁵	—	2	4	no	—
95	C₂²×D₄	—	2	32	no	—
96	C₂²×D₄	—	2	12	no	—
97	C₂²×D₄	—	2	48	no	—
98	C₂²×D₄	—	4	32	no	—
99	C₂²×D₄	C₂³×C₁₃	1	24	yes	—
100	C₂³×C₄	D₄×C₁₃	1	2	yes	—
101	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
102	C₂×D₈	—	2	192	no	—
103	C₂×D₈	C₂³×C₁₃	1	64	yes	complemented
104	C₂²×C₈	D₄×C₁₃	1	16	yes	complemented
105	C₂²×C₈	C₂³×C₁₃	1	12	yes	—
106	#127	—	2	48	no	—
107	#127	C₂³×C₁₃	1	32	yes	complemented
108	C₂×C₄²	D₄×C₁₃	1	6	yes	—
109	C₂×C₄²	C₂³×C₁₃	1	6	yes	—
110	C₄×C₈	C₂³×C₁₃	1	16	yes	complemented
111	C₂×C₁₃	—	8	16	no	—
112	C₂×C₁₃	C₂×#493	1	12	yes	direct product factor
113	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
114	C₂×C₁₃	C₂³×D₈	1	2	yes	—
115	C₂⁴	—	2	24	no	—
116	C₂⁴	—	4	16	no	—
117	C₂⁴	—	4	32	no	—
118	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
119	C₂×D₄	—	2	96	no	—
120	C₂×D₄	—	2	24	no	—
121	C₂×D₄	—	2	64	no	—
122	C₂×D₄	—	4	48	no	—
123	C₂×D₄	—	4	192	no	—
124	C₂×D₄	C₂⁴×C₁₃	1	16	yes	—
125	C₂²×C₄	C₁₃×D₈	1	16	yes	complemented
126	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
127	C₂²×C₄	C₂×D₄×C₁₃	1	8	yes	—
128	C₂²×C₄	C₂×D₄×C₁₃	1	12	yes	—
129	C₂²×C₄	C₂⁴×C₁₃	1	3	yes	—
130	D₈	—	2	128	no	—
131	C₂×C₈	C₂×D₄×C₁₃	1	48	yes	complemented
132	C₂×C₈	C₂⁴×C₁₃	1	8	yes	—
133	C₄²	C₂×D₄×C₁₃	1	12	yes	—
134	C₄²	C₂⁴×C₁₃	1	4	yes	—
135	C₁₃	C₂²×#493	1	1	yes	characteristic, direct product factor, Sylow
136	C₂³	—	2	16	no	—
137	C₂³	—	4	48	no	—
138	C₂³	—	4	96	no	—
139	C₂³	—	8	64	no	—
140	C₂³	C₁₃×#127	1	4	yes	—
141	C₂³	C₂×C₁₃×D₈	1	8	yes	—
142	C₂³	C₂²×D₄×C₁₃	1	3	yes	—
143	D₄	—	2	32	no	—
144	D₄	—	4	48	no	—
145	D₄	—	4	128	no	—
146	C₂×C₄	C₁₃×#127	1	12	yes	—
147	C₂×C₄	C₂×C₁₃×D₈	1	48	yes	complemented
148	C₂×C₄	C₂²×D₄×C₁₃	1	3	yes	—
149	C₂×C₄	C₂²×D₄×C₁₃	1	12	yes	—
150	C₂×C₄	C₂²×D₄×C₁₃	1	8	yes	—
151	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
152	C₈	C₂²×D₄×C₁₃	1	16	yes	complemented
153	C₂²	—	4	32	no	—
154	C₂²	—	4	16	no	—
155	C₂²	—	8	96	no	—
156	C₂²	C₁₃×#493	1	16	yes	direct product factor
157	C₂²	C₂×C₁₃×#127	1	6	yes	—
158	C₂²	C₂²×C₁₃×D₈	1	12	yes	—
159	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
160	C₄	C₂×C₁₃×#127	1	6	yes	—
161	C₄	C₂²×C₁₃×D₈	1	16	yes	complemented
162	C₄	C₂³×D₄×C₁₃	1	2	yes	—
163	C₂	—	8	16	no	—
164	C₂	C₂×C₁₃×#493	1	12	yes	direct product factor
165	C₂	C₂²×C₁₃×#127	1	1	yes	characteristic
166	C₂	C₂³×C₁₃×D₈	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂³×C₁₃×D₈	C₂	semidirect
2. C₂²×C₁₃×#127	C₂	semidirect
3. C₂×C₁₃×#493	C₂	direct (N × H)
4. C₂²×C₄×C₈×C₁₃	C₂	semidirect
8. C₂²×C₁₃×D₈	C₂²	semidirect
11. C₂×C₁₃×#127	C₂²	semidirect
13. C₁₃×#493	C₂²	direct (N × H)
14. C₂×C₄×C₈×C₁₃	C₂²	semidirect
24. C₂×C₁₃×D₈	C₂³	semidirect
25. C₂²×C₈×C₁₃	D₄	semidirect
28. C₁₃×#127	C₂³	semidirect
31. C₄×C₈×C₁₃	C₂³	semidirect
32. C₂²×#493	C₁₃	direct (N × H)
43. C₂²×C₄×C₁₃	D₈	semidirect
49. C₂×C₈×C₁₃	C₂×D₄	semidirect
53. C₂³×D₈	C₂×C₁₃	semidirect
54. C₂²×#127	C₂×C₁₃	semidirect
55. C₂×#493	C₂×C₁₃	direct (N × H)
56. C₂²×C₄×C₈	C₂×C₁₃	semidirect
68. C₂×C₄×C₁₃	C₂×D₈	semidirect
73. C₈×C₁₃	C₂²×D₄	semidirect
77. C₂²×D₈	C₂²×C₁₃	semidirect
80. C₂×#127	C₂²×C₁₃	semidirect
82. #493	C₂²×C₁₃	direct (N × H)
83. C₂×C₄×C₈	C₂²×C₁₃	semidirect
87. C₂²×C₁₃	#493	direct (N × H)
92. C₄×C₁₃	C₂²×D₈	semidirect
103. C₂×D₈	C₂³×C₁₃	semidirect
104. C₂²×C₈	D₄×C₁₃	semidirect
107. #127	C₂³×C₁₃	semidirect
110. C₄×C₈	C₂³×C₁₃	semidirect
112. C₂×C₁₃	C₂×#493	direct (N × H)
125. C₂²×C₄	C₁₃×D₈	semidirect
131. C₂×C₈	C₂×D₄×C₁₃	semidirect
135. C₁₃	C₂²×#493	direct (N × H)
147. C₂×C₄	C₂×C₁₃×D₈	semidirect
152. C₈	C₂²×D₄×C₁₃	semidirect
156. C₂²	C₁₃×#493	direct (N × H)
161. C₄	C₂²×C₁₃×D₈	semidirect
164. C₂	C₂×C₁₃×#493	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₁₃×D₈	2
C₂²×C₁₃×#127	2
C₂×C₁₃×#493	2
C₂²×C₄×C₈×C₁₃	2
C₂²×#493	13

Subgroups — C22×C13×#493 #15746968

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#493 #15746968